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c270adf | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | import OAI.Combinatorics.InfiniteMatroid.CoveringReduction
import OAI.Combinatorics.InfiniteMatroid.PairDuality
import OAI.Combinatorics.InfiniteMatroid.Main
namespace OAI
namespace InfiniteMatroidCounterexample
open Set Matroid
theorem separate_covering_packing_counterexamples :
(∃ (F : Set E) (M : Bool → Matroid E),
(∀ i, (M i).E = F) ∧ ¬ CoveringConjectureFor F M) ∧
(∃ (F : Set E) (M : Bool → Matroid E),
(∀ i, (M i).E = F) ∧ ¬ PackingConjectureFor F M) := by
obtain ⟨_, _, M₀, M₁, hE₀, hE₁, _, _, _, hno⟩ := main
let M : Bool → Matroid E := fun i ↦ if i then M₁ else M₀
have hg : ∀ i, (M i).E = univ := by
intro i
cases i <;> simp [M, hE₀, hE₁]
have hn : ¬ HasPackingCovering (M false) (M true) := hno
obtain ⟨C, N, hN, hnc, hcond⟩ := separate_covering_of_no_packingCovering hg hn
refine ⟨⟨C, N, hN, fun h ↦ hnc (h.mpr hcond)⟩,
⟨C, fun i ↦ (N i).dual, ?_, ?_⟩⟩
· simpa only [Matroid.dual_ground] using hN
· exact separate_packing_of_covering_counterexample hN hnc hcond
end InfiniteMatroidCounterexample
end OAI
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